


The Chinese Remainder Theorem

by eve11



Category: Doctor Who, Doctor Who (1963), Doctor Who (Big Finish Audio)
Genre: Gen, Mathematics, One Shot, Vignette
Language: English
Status: Completed
Published: 2011-06-12
Updated: 2011-06-12
Packaged: 2017-10-20 08:55:03
Rating: General Audiences
Warnings: No Archive Warnings Apply
Chapters: 1
Words: 1,527
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/210980
Author URL: https://archiveofourown.org/users/eve11/pseuds/eve11
Summary: <blockquote class="userstuff">
              <p>A quiet moment between Six and Evelyn, with kites and modular arithmetic.</p>
            </blockquote>





	The Chinese Remainder Theorem

"Did I ever tell you about the time Sun Tzu told me to go fly a kite?" The Doctor presses a finger down on the cat's cradle of taut string in his hand, like he's striking a piano key, and the multicoloured splash in the sky dances a distant circle. He takes a few steps out onto the grass, a multicoloured splash himself, letting the kite swirl and twirl, and then turns around with an indicative eyebrow and an immodest grin.

Evelyn stretches out in her creaky rattan chair. If she were thirty-five years younger she'd be sprawled on the blanket on the ground, perhaps propped up by her elbows to meet the Doctor's coy smile with one of her own. Instead she does her best to ignore the fact of the chair and the protests of her joints, and points her toes in the sunlight. "Doctor, you're changing the subject."

"I never change the subject."

Her smile is still there, emerging from under a floppy hat and a large pair of sunglasses. "Of course you don't. But we were talking about regeneration, whatever that is."

"So we were." The kite catches his attention with a nosedive and he coaxes it back up with a silent minuet. Evelyn eyes the pile of kites at the edge of the blanket. The wind seems to be picking up, but her knees are still aching from that near escape on Zevin III, and besides, they've got the whole afternoon ahead of them.

"And what are we discussing now?" she asks.

"It was a compliment in that era, to be told to go fly a kite, you know. Not just anyone can boast about a great Chinese general suggesting he collect some sticks and cloth and throw them to the wind."

"Doctor--"

"Well, don't you want to know why he bestowed such an honour upon me?"

Evelyn starts to answer, but the breeze tugs at her hat and instead she claps a hand down on her head, pressing the brim against her cheek. It's a warm wind, the kind that makes the sun bearable, but it's not so good for straw headwear. Evelyn is more than a little dismayed that she may have to head back to the wardrobe room to find a suitable replacement, but even as the thought crosses her mind, she feels a lump of something on her head under her hand.

"Go on," she says, taking off the hat and examining it. Two coils of ribbon are now nestled inside, attached at opposite sides by the brim. She glances back at the TARDIS but it's as mute as always, the blue box unnaturally still, as though it's anchored to something much more solid than the breezy hilltop or the spinning planet beneath it.

"I taught Sun Tzu how to count his armies." The Doctor's coattails flap in the breeze as Evelyn ties down her hat. She lets out a grumpy sigh, more for effect than a sign of actual annoyance.

"We were talking about learning experiences. I asked what taught you the most in life, and with no hesitation you said, 'regeneration.' I asked you what that was, and suddenly we're talking about kites and Chinese generals--"

"Evelyn, for heaven's sake, it's a metaphor!" the Doctor blusters. At his sudden movement, the kite leaps a degree nearer the sun in the sky, the colours washing out behind the glare, the shape wavering. Evelyn just laughs.

"I figured that part out, Doctor, but if I couldn't call you on meandering metaphors from time to time, we'd hardly ever have a meaningful conversation."

"Modular arithmetic," the Doctor says, coaxing the kite back down toward the horizon.

Evelyn sits forward. "Regeneration is modular arithmetic?"

"Sun Tzu counted his armies using modular arithmetic. At least, after I was through with him, anyway."

"And what, exactly, is modular arithmetic?"

The Doctor takes his eyes off the kite, giving Evelyn a quick glance over his shoulder. "Honestly, what do they teach at that university of yours?"

"I'm an historian, not a mathematician. You don't see me assuming that you know . . . " She pauses, trying to think of an example that won't lead to an immediate oral dissertation complete with personal anecdotes.

"Well?" the Doctor asks.

"Oh, never mind."

"Here's a mathematical and historical lesson for you, then," the Doctor says. "Suppose you have a vast collection of things--"

"Armies?"

"People, if you like--a chaotic system generally under your broad control, and you want to learn more about them as a group. To sum them up, perhaps. What would you do?"

"Count them, I suppose."

The Doctor arches an eyebrow. "Well, yes, but how?"

"Line them up in ranks and take a census?"

"Mmm, typical."

"And what's wrong with that?"

"Oh nothing, nothing. Just, well, it's very . . . linear," the Doctor remarks, his eyes drawn again to the speck in the sky.

"Only you could make that sound like an insult," Evelyn says with a mock pout.

The kite makes a few zig-zags just above the treeline, then climbs several degrees and starts in on figure eights.

"How long would it take," the Doctor continues, "to line everyone up in ranks, then to count each person, or even each rank? If the ranks are very long, you have to expend a lot of time making the arrangements and double-checking the numbers in each one. If they're small, you still have a lot of ranks to count."

"If you could see them all at once--"

"Suppose you don't have a bird's eye view." The kite rockets toward the ground, then gracefully arcs skyward again.

"You have a better idea?"

"Of course. You need only pay attention to the odd ones out, provided you pick your ranks precisely, and use several re-arrangements."

"And this is regeneration?"

"No, I told you, it's modular arithmetic." The Doctor turns, dismantling the kite string until it's just a tether in one hand, and leaves its charge momentarily to the open sky. "Let's start with seventeen." He cups the other hand, megaphone-style, around his mouth and shouts, "Ranks of seventeen!" across the valley.

"Doctor!" Evelyn cries, but the few locals scattered around the valley pay them no heed. She closes her eyes, imagining a vast army splayed out in every direction, slowly arranging itself into groups of seventeen. When she opens her eyes again, the Doctor is beaming.

"Now, I trust there's enough co-ordination among my charges to handle getting into groups of seventeen," he says. "It's not that difficult. And when everything is arranged, just so, let's say I discover that the entire army has amassed itself into groups of seventeen, and there are no stragglers or odd men out."

"Seventeen divides your army evenly."

"We've learned that, indeed." The Doctor frowns. "Not very helpful, is it? Seventeen divides lots of numbers evenly. So let's try a few more re-arrangements. Say, nineteen, twenty-three, twenty-nine, those are--"

"Prime numbers." Evelyn recognizes the progression. "Numbers whose divisors are only themselves and one." At the Doctor's stare, she quirks a smile. "I do work at a university," she teases.

"Right. So, tell me, how many stragglers do I have after forming ranks of nineteen?"

"Hmm?"

"Oh for--" the Doctor harrumphs, and says only slightly condescendingly, "Pick a number, any number, between zero and eighteen."

Evelyn smiles, catching on. "Let's say. . . two."

"Ranks of twenty-three?"

"Four."

"Twenty-nine?"

"Ummm . . . fifteen."

"Thirty-one?"

"Twenty-four."

"Fifty-eight thousand, seven hundred and sixty nine."

"I beg your pardon?"

"That's the exact sum of your army. It's the only number that fits the bill. Well--" the Doctor pauses in brief thought, "it's the only number between one and six million, six hundred and seventy-eight thousand, six hundred and seventy one, at any rate."

"You're pulling my leg," Evelyn says.

"Funny, that's what Sun Tzu said at first, until I showed him the logic of it. But it's actually a simple calculation. In five swift re-arrangements, you've learned more than you could after a lifetime of tedious counting." The Doctor squints up at the sky, a hand on his hip, contemplative for a moment. "And that?" he adds. "That's regeneration."

"Doctor," Evelyn says, "I still don't understand--"

"I never told you about Sun Tzu?" the Doctor interrupts. "Fascinating man, very well-rounded for a general."

Evelyn sighs. Not changing the subject, indeed. "Yes, he told you to go--oh!"

She points toward the sky, and the Doctor follows her gaze. "Blast!" he cries, tugging at the kite string to no avail. The multicoloured speck plummets toward the ground, and the Doctor rushes unreservedly downhill, gathering slack as fast as he can, shouting ultimatums at sticks and cloth as he goes. Evelyn watches with bemused fascination, as it becomes more and more difficult to decide which bright splash of colour is controlling the one at the other end of the tether.

She stands slowly, stretching her arms and collecting a kite from among the pile of canvas and angles. She's still not clear on this whole regeneration concept, but it’s a beautiful day with the wind picking up, and the open sky is calling to her. Perhaps she'll ask the Doctor again, some other time.


End file.
